Question
If \[\left| \begin{array}{l}
x - 4\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,x - 4\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,x - 4\,\,
\end{array} \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2},\] then the ordered pair $$(A, B)$$ is equal to:
A.
$$(- 4, 3)$$
B.
$$(- 4, 5)$$
C.
$$(4, 5)$$
D.
$$(- 4, - 5)$$
Answer :
$$(- 4, 5)$$
Solution :
\[{\rm{Here, }}\left| \begin{array}{l}
x - 4\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,x - 4\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,x - 4
\end{array} \right| = \left( {A + Bx} \right){\left( {x - A} \right)^2}\]
$${\text{Put }}x = 0$$
\[ \Rightarrow \,\,\left| \begin{array}{l}
- 4\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\
\,\,0\,\,\,\,\,\,\, - 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\\
\,\,0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\, - 4
\end{array} \right| = {A^3}\]
$$\eqalign{
& \Rightarrow \,\,{A^3} = {\left( { - 4} \right)^3} \cr
& \Rightarrow \,\,A = - 4 \cr} $$
\[ \Rightarrow \,\,\left| \begin{array}{l}
x - 4\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,x - 4\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,x - 4
\end{array} \right| = \left( {Bx - 4} \right){\left( {x + 4} \right)^2}\]
Now take $$x$$ common from both the sides
\[\therefore \,\,\left| \begin{array}{l}
1 - \frac{4}{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,1 - \frac{4}{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\\
\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2x\,\,\,\,\,\,\,\,\,\,\,\,\,1 - \frac{4}{x}
\end{array} \right| = \left( {B - \frac{4}{x}} \right){\left( {1 + \frac{4}{x}} \right)^2}\]
$${\text{Now take }}x \to \infty ,\,{\text{then }}\frac{1}{x} \to 0$$
\[ \Rightarrow \,\,\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,2\\
2\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,\,2\\
2\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,1
\end{array} \right| = B\]
$$ \Rightarrow \,\,B = 5$$
∴ Ordered pair $$(A, B)$$ is $$(- 4, 5)$$